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Gradient Find - The point the gradient of the tangent at (0 H U verify this yourselll, L 6 marks for this part.Hence - the equation of the tangent atScore: 6/6Lu [...

Question

Gradient Find - The point the gradient of the tangent at (0 H U verify this yourselll, L 6 marks for this part.Hence - the equation of the tangent atScore: 6/6Lu [1 JI

Gradient Find - The point the gradient of the tangent at (0 H U verify this yourselll, L 6 marks for this part. Hence - the equation of the tangent at Score: 6/6 Lu [ 1 JI



Answers

If $f(x, y)=x y,$ find the gradient vector $\nabla f(3,2)$ and use it to find the tangent line to the level curve $f(x, y)=6$ at the point $(3,2)$ . Sketch the level curve, the tangent line, and the gradient vector.

In this problem, you're given these shown information nasty. First sketch the level curve on the graph. And so our level curve IHS f of X Y is equal to six. And so that is X Y is equal to six. And so we're going to want to sketch the all the points where X times wine is equal to six. And so, for example, we know that six is it when someone wise go to six, extent would be equal to one. And so one of the points the level curve is gonna be right here. We also know that when extra people to six, why is equal to one So one of the points is going to be right here. And then we also know that when X is equal to three, y is equal to two. And when why is equal to three X is equal to two. Much is right. Okay. And so now I am going to draw in line connecting these and that is going to be our level curve right there and again, that is Oh, my bad. Here we go. X Y is equal to six and now we want to sketch the tangent line. To do this, we need our partials. And so our personal with respect X is going to be equal to why and our partial with respect to why it's going to be equal to s. And so we're going to evaluate the's at our point given and so why. Evaluated at 32 is equal to two and X evaluated up three To is equal to three. And to find the tangent online we're going to take are partial with respect X and multiply it Times X minus. The X coordinate we were given, which is three and then we're going to add to that, are partial with respect to why times why minus the y value we were given, which is two and said that equal to zero. And when we simplify this, we get two X minus. Six says three wine minus six is equal to zero in 1906 minus six is 12 and that to both sides we get two X plus three y is equal to 12. And so I want to write this as you would normally see a line by moving extroverts the other side. So we get three why equals negative two x plus 12 in dividing both sides by three we get why this sequel to Negative Too Thirds X Plus four. And so when when we grabbed this we know that are why intercept is equal to four. And we know that we're going down to and why. And over three and X and so down to and why did us two to over three and wind assisted three sore right here, which is the point we want three comma two and then again, down to like it. Just 0/3 gets us to six, and I'm going to draw a line connecting these points, and that is our tangent line. And again, that is the line to expose. Three y equals 12 and now we want to sketch the Grady in Vector. Now our ingredient doctor. Another way that we're going to graph arm Radian Vector is that we're going to make it perpendicular to this point based on the level curve in the tangent line. And so it's going to be roughly in this direction, and we're going to want to go up three and over two, based on our three and two over here. Eso up three reaches us to about five and over to also gets us to about five. And so that is going to be roughly my endpoint five. But I just actually over here, and so I'm going toe cleaning up a little bit. And that is my greedy and defector. And now we found everything we need to find.

We're giving a function think she and a point. And we're asked to find the gradient vector of the function at this point. And to use it to find the tangent line to a level curve at this point, they were asked to sketch the level curve, the tangent line and the gradient vector. So we're giving a function F of x. Y equals X times Y mm. We're asked to find the greeting victor. The greeting of f. at the .32. Well the greening of F is the vector why X. And therefore the greeting of f. at the .32 is the vector two, three. And the show here certain units Marie my pal you're in now we know that obviously. Yeah I'm struck so cute. The equation for the tangent line to the level curve at Quebec's Y equals six at the point 32 is given by the ingredient of F At 3, 2 dotted with the vector x minus three, Y -2 equals zero. And this is two times x minus three. Plus three times y minus two equals zero. They sponsor is sure this simplifies to two X plus three. Y -12 equals zero. Yeah. Imagine resistant to smoke. This is the equation of the tangent line. Now we'll sketch the level curve, the tangent line and the grading vector. So the level curve is in the xy plane. The old yourself. Yeah microbial life to you, judge. And it looks something like this. Pretty sure. Let's see points. Mhm. Yeah. You sick. Yeah college 20% off. So we have our level curve but I'll draw in. Red has points at 16 and 61. It also has points at 2 3 and 32 mm. It looks something like this was off. No the tangent line was two X plus three, Y minus 12 equals zero. This has an X intercept uh X equals six. So 60 Mhm. Mhm. And it has a Y intercept okay of 04 And so the tangent line looks something like this. This is an entirely accurate sketch, but it should really be touching the graph At the .32, which is about here, I dialed the wrong. And finally the gradient vector Is the Victor 2 3. Now position it so that it's starting point is at the .32. Now draw it in blue. The green victor looks something like this, notice that it is perpendicular to the level curve. Yeah, no.

Uh, question X square nickel to the minus six. Why? And upon minus 66 Here we're cone. Abound. Ah, Formula funding hunting line. It would be wise in coaching a wiser plus wife prime X minus X zero. And in this case him. We need to find the white Bram. So why prime in with Nico June? Uh ah. Here we divide him by money six. So we have minus one out of three. Thanks. And then end up on X and Y in here. Doesn't implies. Done. Then why primary with Nico? Attuned. Uh, minus one. Took three times minus six. So we can go to that. You there followed Dunton lie. It will be under form. Why, Nico? Jew wiser it could. Your six plus jewels Thames. Thanks less six We signifying get Why it with Nico? Jude? Uh do thanks. And then last This one will be 18

In this question we want to find equation attention to this Curfew at this point to one. Let's take a look at the basic rules you require when I different shape. Extra power and respect to X. I bring down the power and we beat the eggs and subtract one to his power. Special case will be extra power one. When differentiated with respect to X. I'll get one and are special cases. When a differentiate constant number respect to X. I get zero. No, we're not different ship. The 90 friendship a function of X to the power in with respect to X. Bring down the power and beat the function. Subtract one to its power. And then differentiate the function in some special case will be when a differentiate white or power one where Y is a function of X. I'll just get dy over the next. Now for that rule, if I have two functions of X multiplied to each other. You envy freeze the first function, differentiate the second one. Put a plus, freeze the second function, differentiate the first one. All right, let's take a look at this equation here that's differentiating we respect to X on both sides. Good. So on the left side I have two times added together. So I can do turn by turn differentiation for the first time X. Y. I will require for that room so far east of us function X differentiate the second function. Y. W. Dy over the X. Put the plus. Now I freeze my second function. Why differentiate my first function. X? Excellent. Friendship with respect to X. Or just get one now on to the next one. X square white square Again. Our need for that rule, freeze the first function. X square differentiate my white square. Using this differentiating wide square. Bring down the power to just repeat the wine. Subtract one to the power. That's just one. Now don't forget to differentiate why they would just be Dy over the X. Plus. Now freeze my wife square and differentiate my first function here. X square. So bring on the power to we beat the X. Subject ones runs power and that's it Equals two on my right side is the number. And differentiate that respect and I'll just get zero. Now tidy up group, order the U. N. T. S onto one side and the rest will be on the other side. So I'll get this X plus two X squared Y. Do you? Y over dx is equals two minus Y minus two X. Y square. Now for here I'll just into one. We're X equals to two. White goes to one. So I just said 21 Yeah. Yeah. Yeah. Thank you. Finally my dy dx is minus half so therefore equation of tangent at this point we'll be we'll be using the y minus Y. One equals two. M x minus X one formula. So I have wine minus my white one is one. My gradient M is minus half, which is the DVD extent. This is my name, an X minus my ex one is two, so rearranging it, my final answer would be Why equals 2 -1 x. Last two.


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